Q. A circle is inscribed in a triangle with sides of lengths 7 cm, 8 cm, and 9 cm. What is the radius of the inscribed circle?
A.
4 cm
B.
3 cm
C.
2 cm
D.
5 cm
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Solution
The semi-perimeter s = (7 + 8 + 9)/2 = 12 cm. The area A can be calculated using Heron's formula. The radius r = A/s. The area is 24 cm², so r = 24/12 = 2 cm.
Correct Answer:
B
— 3 cm
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Q. A circle is inscribed in a triangle. If the triangle has sides of lengths 7, 8, and 9 units, what is the radius of the inscribed circle?
A.
3 square units
B.
4 square units
C.
5 square units
D.
6 square units
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Solution
The area of the triangle is 24 square units (using Heron's formula). The semi-perimeter is 12 units. The radius r = Area/semi-perimeter = 24/12 = 2 units.
Correct Answer:
B
— 4 square units
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Q. If the area of a sector of a circle is 20π square units and the radius is 10 units, what is the angle of the sector in radians?
A.
1 radian
B.
2 radians
C.
3 radians
D.
4 radians
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Solution
The area of a sector is given by (θ/2) * r². Thus, 20π = (θ/2) * 10². Solving gives θ = 4 radians.
Correct Answer:
B
— 2 radians
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Q. If the coordinates of the vertices of a triangle are A(1, 2), B(4, 6), and C(1, 6), what is the area of the triangle?
A.
6 square units
B.
8 square units
C.
10 square units
D.
12 square units
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Solution
Using the formula Area = 1/2 |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|, we find Area = 1/2 |1(6 - 6) + 4(6 - 2) + 1(2 - 6)| = 1/2 |0 + 16 - 4| = 1/2 * 12 = 6 square units.
Correct Answer:
A
— 6 square units
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Q. If two circles intersect at points A and B, what can be said about the line segment AB?
A.
It is a diameter of both circles
B.
It is a chord of both circles
C.
It is a tangent to both circles
D.
It is a secant to both circles
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Solution
The line segment AB is a chord of both circles since it connects two points on each circle.
Correct Answer:
B
— It is a chord of both circles
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Q. In a right triangle, if one angle is 30 degrees and the hypotenuse is 10 units, what is the length of the side opposite the 30-degree angle?
A.
5 units
B.
10 units
C.
√3 units
D.
√2 units
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Solution
In a 30-60-90 triangle, the side opposite the 30-degree angle is half the hypotenuse. Thus, it is 10/2 = 5 units.
Correct Answer:
A
— 5 units
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Q. In triangle ABC, if angle A = 90 degrees, angle B = 45 degrees, what is the measure of angle C?
A.
45 degrees
B.
60 degrees
C.
30 degrees
D.
90 degrees
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Solution
The sum of angles in a triangle is 180 degrees. Therefore, angle C = 180 - 90 - 45 = 45 degrees.
Correct Answer:
A
— 45 degrees
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Q. What is the area of a circle with a diameter of 12 units?
A.
36π square units
B.
144π square units
C.
24π square units
D.
48π square units
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Solution
The radius is 6 units (diameter/2). Area = πr² = π(6)² = 36π square units.
Correct Answer:
A
— 36π square units
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Q. What is the length of the arc of a circle with a radius of 4 units and a central angle of 90 degrees?
A.
2π units
B.
4π units
C.
π units
D.
8 units
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Solution
Arc length = (θ/360) * 2πr = (90/360) * 2π(4) = (1/4) * 8π = 2π units.
Correct Answer:
A
— 2π units
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Q. What is the radius of a circle with a circumference of 31.4 units?
A.
5 units
B.
10 units
C.
15 units
D.
20 units
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Solution
The formula for circumference is C = 2πr. Solving for r gives r = C/(2π) = 31.4/(2π) = 5 units.
Correct Answer:
A
— 5 units
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